Problem 17 - Fundamentals of Physics

Question

A uniformly charged conducting sphere of \(1.2 \mathrm{~m}\) diameter has surface charge density \(8.1 \mu \mathrm{C} / \mathrm{m}^{2} .\) Find (a) the net charge on the sphere and (b) the total electric flux leaving the surface.

Step-by-Step Solution

Answer

(a) The net charge is approximately \(3.66 \times 10^{-5}\) C. (b) The total electric flux is approximately \(4.14 \times 10^{6}\) N·m²/C.

Step 1: Find the Radius of the Sphere

Given the diameter of the sphere is 1.2 m. To find the radius \( r \), we use the formula:\[ r = \frac{\text{diameter}}{2} = \frac{1.2}{2} = 0.6 \text{ m} \]

Step 2: Calculate the Surface Area of the Sphere

The surface area \( A \) of a sphere is given by:\[ A = 4\pi r^2 \]Plugging in \( r = 0.6 \text{ m} \):\[ A = 4\pi (0.6)^2 = 4\pi \times 0.36 \approx 4.52 \text{ m}^2 \]

Step 3: Calculate the Net Charge on the Sphere

The net charge \( Q \) can be calculated using the surface charge density \( \sigma \) and the surface area \( A \):\[ Q = \sigma \times A \]Given \( \sigma = 8.1 \mu \text{C/m}^2 = 8.1 \times 10^{-6} \text{ C/m}^2 \):\[ Q = 8.1 \times 10^{-6} \times 4.52 \approx 3.66 \times 10^{-5} \text{ C} \]

Step 4: Calculate the Total Electric Flux Leaving the Surface

According to Gauss's Law, the total electric flux \( \Phi \) leaving the surface is given by:\[ \Phi = \frac{Q}{\varepsilon_0} \]where \( \varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2 \) is the permittivity of free space and \( Q = 3.66 \times 10^{-5} \text{ C} \):\[ \Phi = \frac{3.66 \times 10^{-5}}{8.85 \times 10^{-12}} \approx 4.14 \times 10^{6} \text{ N}\cdot\text{m}^2/\text{C} \]

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux passing through a closed surface to the charge enclosed by that surface. It states that the total electric flux through a closed surface, also known as a Gaussian surface, is equal to the net charge enclosed divided by the permittivity of free space. The equation for Gauss's Law is:\[ \Phi = \frac{Q}{\varepsilon_0} \]where:

• \( \Phi \) is the total electric flux through the surface.
• \( Q \) is the net charge within the enclosed surface.
• \( \varepsilon_0 \) is the permittivity of free space, approximately equal to \( 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2 \).

Understanding Gauss's Law is crucial because it allows the calculation of electric fields from symmetric charge distributions, simplifying complex problems. It shows how charged particles generate an electric field that affects the distribution of charge in surrounding materials. Students can utilize Gauss's Law when dealing with problems involving spherical, cylindrical, or planar symmetry, as it provides a powerful tool for calculating electric fields without directly integrating over the entire charge distribution.

Surface Charge Density

Surface charge density is a measure of how much electric charge is accumulated per unit area on a surface. It is usually denoted by \( \sigma \) and is measured in units of C/m² (Coulombs per square meter). The surface charge density can be calculated using the formula:\[ \sigma = \frac{Q}{A} \]where:

• \( \sigma \) is the surface charge density.
• \( Q \) is the total charge distributed over the surface.
• \( A \) is the area over which the charge is distributed.

Surface charge density provides insight into how charges are spread over a surface and can affect the behavior of electric fields around conductors. In the context of a uniformly charged conducting sphere, the surface charge density is constant over the entire sphere, indicating that the charge is distributed evenly. This uniform distribution simplifies the calculations of electric fields and electric potential, making it an essential concept in electrostatics.

Uniformly Charged Conducting Sphere

A uniformly charged conducting sphere is a sphere that has charge distributed evenly across its surface. Conductors differ from insulators in that the charges within them move freely, reaching a state where the surface charge density becomes uniform due to electrostatic repulsion. The electric field inside a charged conducting sphere is zero due to this redistribution of charges. For a sphere with radius \( r \) and a surface charge density \( \sigma \), the total charge \( Q \) can be found using the formula:\[ Q = \sigma \times A \]where:

• \( A = 4\pi r^2 \) is the surface area of the sphere.
• \( Q \) is the total net charge on the sphere.

In electrostatic equilibrium, the electric field outside the sphere behaves as if all the charge were concentrated at the center of the sphere, allowing us to easily calculate the electric field and use Gauss's Law effectively. Understanding the behavior of a uniformly charged conducting sphere is fundamental in solving problems related to electrostatics, as it exemplifies how charges distribute in conductive materials.